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Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem

Abstract

We use reflecting Brownian motion (RBM) to prove the well-known Gauss–Bonnet–Chern theorem for a compact Riemannian manifold with boundary. The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.

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References

  1. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin (1992)

    Book  Google Scholar 

  2. Bismut, J.M.: Index theorem and the heat equation. In: Proceedings of the International Congress of Mathematicians, vol. 2. American Mathematical Society (1987)

  3. Bruning, J.: Ma, Xiaonan: An anomaly formula for Ray-Singer metrics on manifolds with boundary. Geom. Fund. Anal. (GAFA) 45, 767–837 (2006)

    Article  Google Scholar 

  4. Bruning, J.: Ma, Xiaonan: On the gluing formula for the analytic torsion. Math. Z. 273, 1085–1117 (2013)

    MathSciNet  Article  Google Scholar 

  5. Chern, S.S.: A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds. Ann. Math. 45, 741–752 (1944)

    MathSciNet  Article  Google Scholar 

  6. Duff, G.F.D.: Differential forms in manifolds with boundary. Ann. Math. 56(1), 115–127 (1952)

    MathSciNet  Article  Google Scholar 

  7. Getzler, E.: A short proof of the Atiyha–Singer index theorem. Topology 25, 111–117 (1986)

    MathSciNet  Article  Google Scholar 

  8. Gilkey, P.B.: lnvariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, 2nd edn. CRC Press, Boca Raton (1994)

    Google Scholar 

  9. Gilkey, P.B.: The boundary integrand in the formula for the signature and Euler characteristic of a Riemannian manifold with boundary. Adv. Math. 15, 334–360 (1975)

    MathSciNet  Article  Google Scholar 

  10. Hsu, E.P.: Short-time asymptotics of the heat kernel on a concave boundary. SIAM J. Math. Anal. 20(5), 1109–1127 (1989)

    MathSciNet  Article  Google Scholar 

  11. Hsu, E.P.: Stochastic Analysis on Manifolds. Graduate Studies in Mathematics, vol. 38. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  12. Hsu, E.P.: On the principle of not feeling the boundary. J. Lond. Math. Soc. 2(51), 373–382 (1995)

    MathSciNet  Article  Google Scholar 

  13. Hsu, E.P.: On the $\Theta $-function of a Riemannian manifold with boundary. Trans. Am. Math. Soc. 333(2), 643–671 (1992)

    MathSciNet  MATH  Google Scholar 

  14. Hsu, E.P.: Multiplicative functional for the heat equation on manifolds with boundary. Mich. Math. J. 50(2), 351–367 (2002)

    MathSciNet  Article  Google Scholar 

  15. Hsu, E.P.: Stochastic Local Gauss–Bonnet–Chern Theorem. J. Theor. Probab. 10, 819–834 (1997)

    MathSciNet  Article  Google Scholar 

  16. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland (1984)

  17. McKean, H.P., Jr., Singer, I.: Curvature and eigenvalues of the Laplacian. J. Differ. Geom. 1, 43–69 (1967)

    MathSciNet  Article  Google Scholar 

  18. Norris, J.R.: Path integral formulae for heat kernels and their derivatives. Probab. Theory Relat. Fields 94, 525–541 (1993)

    MathSciNet  Article  Google Scholar 

  19. Ouyang, C.: Multiplicative functionals for the heat equation on manifolds with boundary. In: Stochastic Analysis and Related Topics—A Festschrift in Honor of Rodrigo Banuelos, Progress in Probability, vol. 72, pp. 67–83 . Birkhauser (2017)

  20. Patodi, V.P.: Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 5, 251–283 (1971)

    MathSciNet  MATH  Google Scholar 

  21. Shigekawa, I., Ueki, N., Watanabe, S.: A probabilistic proof of the Gauss–Bonnet–Chern theorem for manifolds with boundary. Osaka J. Math. 26(4), 897–930 (1989)

    MathSciNet  MATH  Google Scholar 

  22. Yu, Y.: The Index Theorem and the Heat Equation Method. World Scientific, Singapore (2001)

    Book  Google Scholar 

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Acknowledgements

The second author would like to thank Professor Denis Bell of the University of North Florida for his interest in the project and helpful discussions during the early stage of the investigation. The authors also thank Professor Xiaonan Ma of Universite Paris 7 for drawing their attention to the papers [3] and [4].

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Correspondence to Weitao Du.

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Du, W., Hsu, E.P. Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem. Commun. Math. Stat. (2022). https://doi.org/10.1007/s40304-021-00266-3

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  • DOI: https://doi.org/10.1007/s40304-021-00266-3

Keywords

  • Manifold with boundary
  • Gauss–Bonnet–Chern theorem
  • Reflecting Brownian motion

Mathematics Subject Classification

  • 60J65
  • 58J65